Average Error: 33.1 → 9.2
Time: 51.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5244621186076104 \cdot 10^{+55}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.029020104220027 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.35488256262753 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.5244621186076104 \cdot 10^{+55}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.029020104220027 \cdot 10^{-124}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 2.35488256262753 \cdot 10^{+48}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\end{array}
double f(double a, double b_2, double c) {
        double r955484 = b_2;
        double r955485 = -r955484;
        double r955486 = r955484 * r955484;
        double r955487 = a;
        double r955488 = c;
        double r955489 = r955487 * r955488;
        double r955490 = r955486 - r955489;
        double r955491 = sqrt(r955490);
        double r955492 = r955485 - r955491;
        double r955493 = r955492 / r955487;
        return r955493;
}
↓
double f(double a, double b_2, double c) {
        double r955494 = b_2;
        double r955495 = -1.5244621186076104e+55;
        bool r955496 = r955494 <= r955495;
        double r955497 = -0.5;
        double r955498 = c;
        double r955499 = r955498 / r955494;
        double r955500 = r955497 * r955499;
        double r955501 = -6.029020104220027e-124;
        bool r955502 = r955494 <= r955501;
        double r955503 = a;
        double r955504 = r955498 * r955503;
        double r955505 = r955494 * r955494;
        double r955506 = r955505 - r955504;
        double r955507 = sqrt(r955506);
        double r955508 = r955507 - r955494;
        double r955509 = r955504 / r955508;
        double r955510 = r955509 / r955503;
        double r955511 = 2.35488256262753e+48;
        bool r955512 = r955494 <= r955511;
        double r955513 = 1.0;
        double r955514 = -r955494;
        double r955515 = r955514 - r955507;
        double r955516 = r955503 / r955515;
        double r955517 = r955513 / r955516;
        double r955518 = 0.5;
        double r955519 = r955518 * r955499;
        double r955520 = r955494 / r955503;
        double r955521 = 2.0;
        double r955522 = r955520 * r955521;
        double r955523 = r955519 - r955522;
        double r955524 = r955512 ? r955517 : r955523;
        double r955525 = r955502 ? r955510 : r955524;
        double r955526 = r955496 ? r955500 : r955525;
        return r955526;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.5244621186076104e+55

    1. Initial program 56.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 4.1

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.5244621186076104e+55 < b_2 < -6.029020104220027e-124

    1. Initial program 38.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--38.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified15.5

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified15.5

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if -6.029020104220027e-124 < b_2 < 2.35488256262753e+48

    1. Initial program 12.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num12.6

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    if 2.35488256262753e+48 < b_2

    1. Initial program 35.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 5.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5244621186076104 \cdot 10^{+55}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.029020104220027 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.35488256262753 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))